Natural Theories

TL;DR

The paper develops a rigorous formalism of natural equations and natural theories, grounding general covariance in functorial relations between categories of smooth manifolds and fiber bundles. It introduces naturalization to capture how equations may depend on background structure, clarifies minimal coupling as a first-order, background-dependent notion, and distinguishes three senses of symmetry (diffeomorphism-invariance, spacetime symmetry, and dynamical symmetry) within this framework. A central result shows that no sufficiently rich natural theory can have a well-posed initial value problem, illustrating a fundamental tension between naturality and determinism (the hole argument) and linking it to the necessity of fixing background structure to recover determinism in theories like Maxwell’s and general relativity. The work offers a structured approach to analyzing background dependence, coupling, and symmetry in classical field theories, and it reframes longstanding philosophical debates about covariance and determinism in precise mathematical terms.

Abstract

We consider the class of physical theories whose dynamics are given by natural equations, which are partial differential equations determined by a functor from the category of n-manifolds, for some n, to the category of fiber bundles, satisfying certain further conditions. We show how the theory of natural equations clarifies several important foundational issues, including the status and meaning of minimal coupling, symmetries of theories, and background structure. We also state and prove a fundamental result about the initial value problem for natural equations.

Figures (1)

• Figure 1: The naturality diagrams for natural equations over natural bundles.

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• Theorem 3
• proof